The shaft from problem $3$ is modified by $($i$)$ increasing the diameter from $0\mathrm{.}8$ in to $0\mathrm{.}88$ in$,($ii$)$ adding a groove at section B with a root diameter of $0\mathrm{.}8$ in $,$ and $($iii$)$ reducing $F_B$ from $280$ lb to $160$ lb $.$ The other loads, material properties, surface finish, etc., are unchanged from problem $3.($a$)$ Sketch maximum$-$stress, alternating$-$stress, and mean$-$stress stress elements for the critical section$/$point on the shaft $($show the stresses symbolically$).$ Calculate the static factor of safety using $($b$)$ the MSSTF and $($c$)$ the DETF $($local yielding is permissible, so theoretical stress concentration factors can be ignored for static calculations$).$ Calculate the infinite life fatigue factor of safety at the critical section using $($d$)$ the Soderberg criterion and $($e$)$ the Goodman criterion.

$($a$)$

$T_{x\text{m}\text{a}\text{x}}=1000lb$

$T_{x\text{m}\text{i}\text{n}}=1000lb$